Vjhiker, you will fly from Arizona to a major airport in Europe. From there, yes, you take either another flight or ground transportation to your starting point on the camino.
Start by figuring your best European entry point. Most common are either Madrid or Paris, with London and Barcelona popular as well. Check their relative merits in terms of schedule and price. Then check into the logistics of getting from your arrival point to the Camino. As mentioned above, Rome2Rio is very helpful for this.
Using myself as an example, from southeast Pennsylvania, I can fly non-stop from Philadelphia to Madrid. There are cheaper flights (some much cheaper!) from NYC, but they involve layovers (sometimes involving a change of airports), plane changes, and getting to New York, so for me convenience wins. Others have different critetia. So that's why no one can say "This is the best."
The same principal holds true for getting from your arrival point to the Camino. Check out your various options from each of the arrival cities and then decide your route based on the overall picture. Search through the forum. Many people describe the various options they chose, and almost always make it sound fun. Because getting there is all part of the experience.
As for medical insurance, you definitely want to be covered. If your own policy doesn't cover you overseas, check out travel insurance. It costs very little, as you're not covering cancellation other than your flight cost. But in the unlikely event that you should need it, it could save thousands of dollars.
Happy planning!
